Neuromorphic computing is an emerging computing technology inspired by the operational principles of the human brain. By employing neuromorphic devices to emulate neuronal functions and construct neural networks, it effectively overcomes the von-Neumann bottleneck, demonstrating remarkable energy efficiency. Due to their unique electrical properties, compact structures, and extremely low power consumption, memristive devices have become a core focus in neuromorphic computing based on memristive devices have become a core focus in neuromorphic device research. This review highlights recent advances in neuromorphic computing based on memristive devices, covering the working principles, key functions, and performance metrics of memristive devices, the structure and typical applications of memristive arrays, and the architectures and performance of neuromorphic chips. It also analyzes the intrinsic links among memristive devices, memristive arrays, and neuromorphic chips.